Petri Juutinen

A tour of the theory of absolutely minimizing functions

A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation

On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation

We discuss and compare various notions of weak solution for the p-Laplace equation -\text{div}(|\nabla u|^{p-2}\nabla u)=0 and its parabolic counterpart u_t-\text{div}(|\nabla u|^{p-2}\nabla u)=0. In addition to the usual Sobolev weak solutions based on integration by parts, we consider the p-superharmonic (or p-superparabolic) functions from nonlinear potential theory and the viscosity solutions based on generalized pointwise derivatives (jets). Our main result states that in both the elliptic and the parabolic case, the viscosity supersolutions coincide with the potential-theoretic supersolutions.

The

We consider the non-nonlinear optimal transportation problem of minimizing the cost functional

\infty

\mathcal{C}_\infty(\lambda) = \operatornamewithlimits{\lambda-ess\,sup}_{(x,y) \in \Omega^2} |y-x|

-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps

in the set of probability measures on

A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation

\Omega^2

Convex functions on Carnot groups

having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

On the definition of viscosity solutions for parabolic equations

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation −div(|Du| p−2 Du) = 0 coincide. Our proof is more direct and transparent than the original proof of Juutinen et al. [8], which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation.

Large solutions for the infinity Laplacian

We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.

The p-Laplace eigenvalue problem as p goes to infinity in a Finsler metric

In this short note we suggest a refinement for the definition of viscosity solutions for parabolic equations. The new version of the definition is equivalent to the usual one and it better adapts to the properties of parabolic equations. The basic idea is to determine the admissibility of a test function based on its behavior prior to the given moment of time and ignore what happens at times after that.